A few months, I was involved with conducting mock interviews with one of the local state schools. This was to give them practice and feedback - they also had CV-writing workshops and so on. My job was to interview some of them for about 12 minutes each and give feedback on their clothing, body language, answers (did they answer the question asked, did they expand on their answers, did they have a varied tone of voice and so on.)
In a real job application scenario, many of them wouldn't have made it as far as interview, so it was a far greater range of potential than you'd usually get at that point of an application process.
Some of them were pretty much monosyllabic, even when you prompted them to give an example or expand on things. Others were great - didn't need prompting, gave great, relevant answers which illustrated their experience of what was being asked. With one exception, these were people who were involved with a local sports team, or air cadets or scouts or DofE - that's where they had gained experience, leadership, a sense of responsibility. So do those outside activities - they do make a difference!
The exception was a boy who was a young carer - his single-parent mother has a degenerative disease, and his two siblings are quite a bit younger than he is, and that's where he's developed his time-management and sense of responsibility. He was great, and I really hope he gets the support he needs to do well.
I think those who go to private school are just more likely to get opportunities to lead and take part in extra activities, inside and outside side of school. They're more likely to see people like them doing it - "if you can see it, you can be it." There are state schools which try to do this too, but mostly, they simply don't have the resources to offer a wide enough range of activities that all children can find their niche, and not all parents have the resources or interstitial in helping their child take part in things out of school.
Plus private schools are selective, so quite probably, the disengaged, monosyllabic ones who prove to be resistant to efforts to change that, will have been advised that they will probably get on better in a different environment - which is an option state schools don't have as easily (despite the numbers getting permanently excluded.)
And the other thing is that we will mostly hear from the confident ones. The ones who end up full of anxiety, fearing failure so not risking failure - they aren't the ones leading businesses and talking on TV or being the ones everyone remembers at the end of a party, because they're disappearing into the background.
Having said that, I still think private education does tend to produce more confident people overall - just not all of them, in the same way that state educated people aren't all lacking confidence either.